A maximum (non-extensive) entropy approach to equity options bid–ask spread

106

118

We present the main features of the mathematical theory generated by the κ-deformed exponential function exp κ (x) = (developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The κ-mathematics has its roots in special relativity and furnishes the theoretical foundations of the κ-statistical mechanics predicting power law tailed statistical distributions, which have been observed experimentally in many physical, natural and artificial systems. After introducing the κ-algebra, we present the associated κ-differential and κ-integral calculus. Then, we obtain the corresponding κ-exponential and κ-logarithm functions and give the κ-version of the main functions of the ordinary mathematics.

Section: Introductionmentioning

confidence: 99%

Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions

Kaniadakis

2013

106

118

“…However, they deal with the price ranges differently. Other approaches include the use of a different entropy functional, as in [21], as well as work on the determination of risk neutral densities using approximations by rational functions, as in [22].…”

Section: Preliminaries and Problem Statementmentioning

confidence: 99%

Risk Neutral Measure Determination from Price Ranges: Single Period Market Models

Gzyl

Molina

Horst

2018

Risk neutral measures are defined such that the basic random assets in a portfolio are martingales. Hence, when the market model is complete, valuation of other financial instruments is a relatively straightforward task when those basic random assets constitute their underlying asset.To determine the risk neutral measure, it is assumed that the current prices of the basic assets are known exactly. However, oftentimes all we know about the current price, or that of a derivative having it as underlying, is a bid-ask range. The question then arises as to how to determine the risk neutral measure from that information. We may want to determine risk neutral measures from that information to use it, for example, to price other derivatives on the same asset. In this paper we propose an extended version of the maximum entropy method to carry out that task. This approach provides a novel solution to this problem, which is computationally simple and fast.

“…The entropy pricing theory established by Gulko showed that the distribution of future price changes was the most random, that was, the choice of price change obeyed the maximum entropy method [ 21 , 22 ]. Tapiero further emphasized the degree of uncertainty in statistical information with incomplete states, and modeled the cross-section of options bid-ask spreads with their strikes by maximizing the Kaniadakis entropy [ 23 ]. The above literature clearly shows that the entropy pricing theory can make full use of the known information and that the maximum entropy method has a certain degree of advantage when used in option pricing.…”

Section: Introductionmentioning

confidence: 99%

Pricing Interval European Option with the Principle of Maximum Entropy

Liu

Zhou

Xiong

et al. 2019

This paper develops the interval maximum entropy model for the interval European option valuation by estimating an underlying asset distribution. The refined solution for the model is obtained by the Lagrange multiplier. The particle swarm optimization algorithm is applied to calculate the density function of the underlying asset, which can be utilized to price the Shanghai Stock Exchange (SSE) 50 Exchange Trades Funds (ETF) option of China and the Boeing stock option of the United States. Results show that maximum entropy distribution provides precise estimations for the underlying asset of interval number situations. In this way, we can get the distribution of the underlying assets and apply it to the interval European option pricing in the financial market.